7 Capacitors and Capacitance

Transcription

1 7 Capacitors and Capacitance A capacitor is constructed of two parallel conducting plates separated by an insulating material. Capacitance is a measure of a capacitor s ability to store charge on its plates. A capacitor has capacitance of 1 farad if 1 coulomb of charge is deposited on the plates by potential difference of 1 volt across the plates. A farad is named after Michael Faraday, a nineteenth century English chemist and physicist. The farad is generally too large a measure of capacitance for most practical applications, so the microfarad (10-6 ) or picofarad (10-12 ) is more commonly used. Q C =, where C is the capacitance, Q is the charge, and V is the V voltage Different capacitors for the same voltage across their plates will acquire greater or lesser amounts of charge on their plates, hence the capacitors have greater or lesser capacitance. The dielectric is the insulator of the capacitor. The purpose of the dielectric is to create an electric field to oppose the electric field setup by free charges on the parallel plates. (Di for opposing and electric for electric field ). The schematic symbol for a capacitor is quite simple, being little more than two short, parallel lines (representing the plates) separated by a gap. Wires attach to the respective plates for connection to other components. When a voltage is applied across the two plates of a capacitor, a concentrated field flux is created between them, allowing a significant difference of free electrons (a charge) to develop between the two plates: As the electric field is established by the applied voltage, extra free electrons are forced to collect on the negative conductor, while free electrons are "robbed" from the

2 positive conductor. This differential charge equates to a storage of energy in the capacitor, representing the potential charge of the electrons between the two plates. The greater the difference of electrons on opposing plates of a capacitor, the greater the field flux, and the greater "charge" of energy the capacitor will store. Because capacitors store the potential energy of accumulated electrons in the form of an electric field, they behave quite differently than resistors (which simply dissipate energy in the form of heat) in a circuit. A capacitor's ability to store energy as a function of voltage (potential difference between the two leads) results in a tendency to try to maintain voltage at a constant level. In other words, capacitors tend to resist changes in voltage drop. When voltage across a capacitor is increased or decreased, the capacitor "resists" the change by drawing current from or supplying current to the source of the voltage change, in opposition to the change. To store more energy in a capacitor, the voltage across it must be increased. This means that more electrons must be added to the (-) plate and more taken away from the (+) plate, necessitating a current in that direction. Conversely, to release energy from a capacitor, the voltage across it must be decreased. This means some of the excess electrons on the (-) plate must be returned to the (+) plate, necessitating a current in the other direction. "A charged capacitor tends to stay charged; a discharged capacitor tends to stay discharged." Hypothetically, a capacitor left untouched will indefinitely maintain whatever state of voltage charge that it's been left it. Only an outside source (or drain) of current can alter the voltage charge stored by a perfect capacitor. Practically speaking, however, capacitors will eventually lose their stored voltage charges due to internal leakage paths for electrons to flow from one plate to the other. Depending on the specific type of capacitor, the time it takes for a stored voltage charge to selfdissipate can be a long time (several years with the capacitor sitting on a shelf!). When the voltage across a capacitor is increased, it draws current from the rest of the circuit, acting as a power load. In this condition the capacitor is said to be charging, because there is an increasing amount of energy being stored in its electric field. Note the direction of current with regard to the voltage polarity: Conversely, when the voltage across a capacitor is decreased, the capacitor supplies current to the rest of the circuit, acting as a power source. In this condition the capacitor is said to be discharging. Its store of energy -- held in the electric field -- is decreasing now as energy is released to the rest of the circuit. Note the direction of current with regard to the voltage polarity:

3 If a source of voltage is suddenly applied to an uncharged capacitor (a sudden increase of voltage), the capacitor will draw current from that source, absorbing energy from it, until the capacitor's voltage equals that of the source. Once the capacitor voltage reached this final (charged) state, its current decays to zero. Conversely, if a load resistance is connected to a charged capacitor, the capacitor will supply current to the load, until it has released all its stored energy and its voltage decays to zero. Once the capacitor voltage reaches this final (discharged) state, its current decays to zero. In their ability to be charged and discharged, capacitors can be thought of as acting somewhat like secondary-cell batteries. The choice of insulating material between the plates, as was mentioned before, has a great impact upon how much charge will develop with any given amount of voltage applied across the plates. Because of the role of this insulating material in affecting charge, it has a special name: dielectric. Not all dielectric materials are equal: the extent to which materials inhibit or encourage the formation of electric field flux is called the permittivity of the dielectric. The measure of a capacitor's ability to store energy for a given amount of voltage drop is called capacitance. Not surprisingly, capacitance is also a measure of the intensity of opposition to changes in voltage (exactly how much current it will produce for a given rate of change in voltage). Capacitance is symbolically denoted with a capital "C," and is measured in the unit of the Farad, abbreviated as "F." 7.1 Capacitors and calculus Capacitors do not have a stable "resistance" as conductors do. However, there is a definite mathematical relationship between voltage and current for a capacitor, as follows:

4 The lower-case letter "i" symbolizes instantaneous current, which means the amount of current at a specific point in time. This stands in contrast to constant current or average current (capital letter "I") over an unspecified period of time. The expression "dv/dt" is one borrowed from calculus, meaning the instantaneous rate of voltage change over time, or the rate of change of voltage (volts per second increase or decrease) at a specific point in time, the same specific point in time that the instantaneous current is referenced at. In this equation we see something novel to our experience thus far with electric circuits: the variable of time. When relating the quantities of voltage, current, and resistance to a resistor, it doesn't matter if we're dealing with measurements taken over an unspecified period of time (E=IR; V=IR), or at a specific moment in time (e=ir; v=ir). The same basic formula holds true, because time is irrelevant to voltage, current, and resistance in a component like a resistor. In a capacitor, however, time is an essential variable, because current is related to how rapidly voltage changes over time. 7.2 Factors affecting capacitance There are three basic factors of capacitor construction determining the amount of capacitance created. These factors all dictate capacitance by affecting how much electric field flux (relative difference of electrons between plates) will develop for a given amount of magnetic field force (voltage between the two plates): PLATE AREA: All other factors being equal, greater plate gives greater capacitance; less plate gives less capacitance. Explanation: Larger plate area results in more field flux (charge collected on the plates) for a given field force (voltage across the plates). PLATE SPACING: All other factors being equal, further plate spacing gives less capacitance; closer plate spacing gives greater capacitance. Explanation: Closer spacing results in a greater field force (voltage across the capacitor divided by the distance between the plates), which results in a greater field flux (charge collected on the plates) for any given voltage applied across the plates. DIELECTRIC MATERIAL: All other factors being equal, greater permittivity of the dielectric gives greater capacitance; less permittivity of the dielectric gives less capacitance. Explanation: Although it's complicated to explain, some materials offer less opposition to field flux for a given amount of field force. Materials with a greater permittivity allow for more field flux (offer less opposition), and thus a greater collected charge, for any given amount of field force (applied voltage)

5 "Relative" permittivity means the permittivity of a material, relative to that of a pure vacuum. The greater the number, the greater the permittivity of the material. Glass, for instance, with a relative permittivity of 7, has seven times the permittivity of a pure vacuum, and consequently will allow for the establishment of an electric field flux seven times stronger than that of a vacuum, all other factors being equal. The following is a table listing the relative permittivities (also known as the "dielectric constant") of various common substances: Material Relative permittivity (dielectric constant) ============================================================ Vacuum Air Bakelite to 6.0 Mica, muscovite to 8.7 Glass-bonded mica to 9.3 Porcelain, Steatite Alumina to 10.0 Distilled water Barium-strontium-titanite An approximation of capacitance for any pair of separated conductors can be found with this formula: A capacitor can be made variable rather than fixed in value by varying any of the physical factors determining capacitance. One relatively easy factor to vary in capacitor construction is that of plate area, or more properly, the amount of plate overlap. 7.3 Series and parallel capacitors When capacitors are connected in series, the total capacitance is less than any one of the series capacitors'individual capacitances. If two or more capacitors are connected

6 in series, the overall effect is that of a single (equivalent) capacitor having the sum total of the plate spacings of the individual capacitors. As we've just seen, an increase in plate spacing, with all other factors unchanged, results in decreased capacitance. Thus, the total capacitance is less than any one of the individual capacitors' capacitances. The formula for calculating the series total capacitance is the same form as for calculating parallel resistances: When capacitors are connected in parallel, the total capacitance is the sum of the individual capacitors'capacitances. If two or more capacitors are connected in parallel, the overall effect is that of a single equivalent capacitor having the sum total of the plate areas of the individual capacitors. As we've just seen, an increase in plate area, with all other factors unchanged, results in increased capacitance. Thus, the total capacitance is more than any one of the individual capacitors' capacitances. The formula for calculating the parallel total capacitance is the same form as for calculating series resistances: As you will no doubt notice, this is exactly opposite of the phenomenon exhibited by resistors. With resistors, series connections result in additive values while parallel connections result in diminished values. With capacitors, it's the reverse: parallel connections result in additive values while series connections result in diminished values. 7.4 Practical considerations Capacitors, like all electrical components, have limitations, which must be respected for the sake of reliability and proper circuit operation

7 Working voltage: Since capacitors are nothing more than two conductors separated by an insulator (the dielectric), you must pay attention to the maximum voltage allowed across it. If too much voltage is applied, the "breakdown" rating of the dielectric material may be exceeded, resulting in the capacitor internally short-circuiting. Polarity: Some capacitors are manufactured so they can only tolerate applied voltage in one polarity but not the other. This is due to their construction: the dielectric is a microscopically thin layer if insulation deposited on one of the plates by a DC voltage during manufacture. These are called electrolytic capacitors, and their polarity is clearly marked. Reversing voltage polarity to an electrolytic capacitor may result in the destruction of that super-thin dielectric layer, thus ruining the device. However, the thinness of that dielectric permits extremely high values of capacitance in a relatively small package size. For the same reason, electrolytic capacitors tend to be low in voltage rating as compared with other types of capacitor construction. 7.5 Capacitor transient response Unlike resistors, which respond instantaneously to applied voltage, capacitors react over time as they absorb and release energy. Because capacitors store energy in the form of an electric field, they tend to act like small secondary-cell batteries, being able to store and release electrical energy. A fully discharged capacitor maintains zero volts across its terminals, and a charged capacitor maintains a steady quantity of voltage across its terminals, just like a battery. When capacitors are placed in a circuit with other sources of voltage, they will absorb energy from those sources, just as a secondary-cell battery will become charged as a result of being connected to a generator. A fully discharged capacitor, having a terminal voltage of zero, will initially act as a short-circuit when attached to a source of voltage, drawing maximum current as it begins to build a charge. Over time, the capacitor's terminal voltage rises to meet the applied voltage from the source, and the current through the capacitor decreases correspondingly. Once the capacitor has reached the full voltage of the source, it will stop drawing current from it, and behave essentially as an open-circuit. When the switch is first closed, the voltage across the capacitor (which we were told was fully discharged) is zero volts; thus, it first behaves as though it were a shortcircuit. Over time, the capacitor voltage will rise to equal battery voltage, ending in a

8 condition where the capacitor behaves as an open-circuit. Current through the circuit is determined by the difference in voltage between the battery and the capacitor, divided by the resistance of 10 k. As the capacitor voltage approaches the battery voltage, the current approaches zero. Once the capacitor voltage has reached 15 volts, the current will be exactly zero. Let's see how this works using real values: The capacitor voltage's approach to 15 volts and the current's approach to zero over time is what a mathematician would call asymptotic: that is, they both approach their final values, getting closer and closer over time, but never exactly reaches their destinations. For all practical purposes, though, we can say that the capacitor voltage will eventually reach 15 volts and that the current will eventually equal zero. REVIEW: A fully discharged capacitor initially acts as a short circuit (current with no voltage drop) when faced with the sudden application of voltage. After charging fully to that level of voltage, it acts as an open circuit (voltage drop with no current). In a resistor-capacitor charging circuit, capacitor voltage goes from nothing to full source voltage while current goes from maximum to zero, both variables changing most rapidly at first, approaching their final values slower and slower as time goes on

9 7.6 Voltage and current calculations There's a sure way to calculate any of the values in a reactive DC circuit over time. The first step is to identify the starting and final voltage values for the capacitor. When the switch in a circuit is closed (or opened), the reactive component will attempt to maintain the voltage at the same level as it was before the switch transition, so that value is to be used for the "starting" value. The final value for voltage is whatever that quantity will be after an infinite amount of time. This can be determined by analyzing a capacitive circuit as though the capacitor was an open-circuit. The next step is to calculate the time constant of the circuit: the amount of time it takes for voltage or current values to change approximately 63 percent from their starting values to their final values in a transient situation. In a series RC circuit, the time constant is equal to the total resistance in ohms multiplied by the total capacitance in farads. The time constant is expressed in units of seconds and symbolized by the Greek letter "tau" (): 7.7 Complex circuits What do we do if we come across a circuit more complex than the simple series configurations we've seen so far? Take this circuit as an example: The simple time constant formula (=RC) is based on a simple series resistance connected to the capacitor. So, what can we do in a situation like this, where resistors are connected in a series-parallel fashion with the capacitor? The answer comes from our studies in network analysis. Thevenin's Theorem tells us that we can reduce any linear circuit to an equivalent of one voltage source, one series resistance, and a load component through a couple of simple steps. To apply Thevenin's Theorem to our scenario here, we'll regard the reactive component (in the

10 above example circuit, the capacitor) as the load and remove it temporarily from the circuit to find the Thevenin voltage and Thevenin resistance. Then, once we've determined the Thevenin equivalent circuit values, we'll re-connect the capacitor and solve for values of voltage or current over time as we've been doing so far. After identifying the capacitor as the "load," we remove it from the circuit and solve for voltage across the load terminals (assuming, of course, that the switch is closed): This step of the analysis tells us that the voltage across the load terminals (same as that across resistor R 2 ) will be volts with no load connected. With a little reflection, it should be clear that this will be our final voltage across the capacitor, seeing as how a fully-charged capacitor acts like an open circuit, drawing zero current. We will use this voltage value for our Thevenin equivalent circuit source voltage. Now, to solve for our Thevenin resistance, we need to eliminate all power sources in the original circuit and calculate resistance as seen from the load terminals: Re-drawing our circuit as a Thevenin equivalent, we get this: Our time constant for this circuit will be equal to the Thevenin resistance times the capacitance (=RC). With the above values, we calculate:

11 Now, we can solve for voltage across the capacitor directly with our universal time constant formula. Let's calculate for a value of 60 milliseconds. Because this is a capacitive formula, we'll set our calculations up for voltage: Again, because our starting value for capacitor voltage was assumed to be zero, the actual voltage across the capacitor at 60 milliseconds is equal to the amount of voltage change from zero, or volts. We could go a step further and demonstrate the equivalence of the Thevenin RC circuit and the original circuit through computer analysis. 7.8 Solving for unknown time Sometimes it is necessary to determine the length of time that a reactive circuit will take to reach a predetermined value. This is especially true in cases where we're designing an RC circuit to perform a precise timing function. To calculate this, we need to modify our "Universal time constant formula." The original formula looks like this: However, we want to solve for time, not the amount of change. To do this, we algebraically manipulate the formula so that time is all by itself on one side of the equal sign, with all the rest on the other side: The ln designation just to the right of the time constant term is the natural logarithm function: the exact reverse of taking the power of e. In fact, the two functions (powers of e and natural logarithms) can be related as such: If e x = a, then ln a = x. If e x = a, then the natural logarithm of a will give you x: the power that e must be was raised to in order to produce a

12 In a resistor-capacitor circuit, we can work "backwards" from previously determined values of voltage to find how long it took to get there. The time constant is 1 second (10 k times 100 µf), and the starting/final values are E C = 0 volts starting and 15 volts final. The calculations below show that it will take 2 seconds for the circuit to reach volts